Page 226 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 226
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In general L and R need not be s-congruences. However:
Lemma 6.2.10. L and R are both s-congruences on a s-band S if and only if for all a,
b, c ∈S,
i) a(bc – cbc)a = a(bc – cbc).
ii) a(bc – bcb)a = (bc – bcb)a.
Proof. Given uLv, (usx)(vsx) = (u + x + xu – uxu – xux)(v + x + xv – vxv – xvx). Distributing
each term of the left factor over the right factor, then adding and simplifying gives
(u) + (xv + x – xvx) + (xu) – (uxu) – (xu) = u + x + xv – uxu – xvx.
(Here we use various consequences of regularity. E.g.,
u(v + x + xv – vxv – xvx) = u + ux + uxv – uxv – uxvx = u + ux – ux = u.)
Thus, (usx)(vsx) = usx holds for uLv if and only if xv – xvx = xu – xux. Replacing u and v by
generic values, uvu and vu, we see that L is a s-congruence if xuvu – xuvux = xvu – xvux holds
for all u, x, v in S. Rearranging the terms gives xvux – xuvux = xvu – xuvu which is equivalent to
(i) upon switching variables. In similar fashion, R being a s-congruence is equivalent to (ii). £
Theorem 6.2.11. Given a s-band S in a ring, both L and R are s-conguences if and
only if s is associative and thus S is a skew lattice.
Proof. If s is associative so that S is a skew lattice, then both L and R must be full skew lattice
congruences. On the other hand, given that L and R are s-congruences, the identities of the
above lemma imply that
a[b, c]2 = a(bc – cbc + cb – bcb) = a(bc – cbc + cb – bcb)a
= a(bc – bcb + cb – cbc)a = (bc – bcb + cb – cbc)a = [b, c]2a.
Thus, the criterion of Theorem 3.2.8 is satisfied and hence s is associative. £
The associativity of s: the role of Kimura factorizations
A closely related criterion involves a canonical factorization that can occur in s-bands.
We begin with bands. Every subsemilattice T of any band S meets each D-class in S in at most
once element. A semilattice section in a band S is a subsemilattice T of S that meets each
D-class exactly once, thus making T is an internal copy of S/D, the maximal semilattice image of
S. If S is also regular, then TR = ∪t∈TRt is a maximal right regular band in S and TL = ∪t∈TLt is a
maximal left regular band in S with TR and TL being copies of the maximal right and left regular
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In general L and R need not be s-congruences. However:
Lemma 6.2.10. L and R are both s-congruences on a s-band S if and only if for all a,
b, c ∈S,
i) a(bc – cbc)a = a(bc – cbc).
ii) a(bc – bcb)a = (bc – bcb)a.
Proof. Given uLv, (usx)(vsx) = (u + x + xu – uxu – xux)(v + x + xv – vxv – xvx). Distributing
each term of the left factor over the right factor, then adding and simplifying gives
(u) + (xv + x – xvx) + (xu) – (uxu) – (xu) = u + x + xv – uxu – xvx.
(Here we use various consequences of regularity. E.g.,
u(v + x + xv – vxv – xvx) = u + ux + uxv – uxv – uxvx = u + ux – ux = u.)
Thus, (usx)(vsx) = usx holds for uLv if and only if xv – xvx = xu – xux. Replacing u and v by
generic values, uvu and vu, we see that L is a s-congruence if xuvu – xuvux = xvu – xvux holds
for all u, x, v in S. Rearranging the terms gives xvux – xuvux = xvu – xuvu which is equivalent to
(i) upon switching variables. In similar fashion, R being a s-congruence is equivalent to (ii). £
Theorem 6.2.11. Given a s-band S in a ring, both L and R are s-conguences if and
only if s is associative and thus S is a skew lattice.
Proof. If s is associative so that S is a skew lattice, then both L and R must be full skew lattice
congruences. On the other hand, given that L and R are s-congruences, the identities of the
above lemma imply that
a[b, c]2 = a(bc – cbc + cb – bcb) = a(bc – cbc + cb – bcb)a
= a(bc – bcb + cb – cbc)a = (bc – bcb + cb – cbc)a = [b, c]2a.
Thus, the criterion of Theorem 3.2.8 is satisfied and hence s is associative. £
The associativity of s: the role of Kimura factorizations
A closely related criterion involves a canonical factorization that can occur in s-bands.
We begin with bands. Every subsemilattice T of any band S meets each D-class in S in at most
once element. A semilattice section in a band S is a subsemilattice T of S that meets each
D-class exactly once, thus making T is an internal copy of S/D, the maximal semilattice image of
S. If S is also regular, then TR = ∪t∈TRt is a maximal right regular band in S and TL = ∪t∈TLt is a
maximal left regular band in S with TR and TL being copies of the maximal right and left regular
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